The upper right-hand corner of a piece of paper, 12 in. by 8 in., as in the figure, is folded over to the bottom edge. How would you fold it so as to minimize the length of the fold? In other words, how would you choose x to minimize y?

I must be missing something. Clearly the minimum length of fold is achieved when the paper is folded in half, so the upper right corner touches the lower right corner. Then the fold is straight across, and is 8" long. Any diagonal fold will be longer.

Of course, I have no diagram, so I don't know whether the paper is portrait or landscape orientation.

To minimize the length of the fold (y), we need to find the optimal value for x.

Let's visualize the situation. We have a rectangular piece of paper with dimensions 12 in. by 8 in. The upper right-hand corner, which we want to fold, gives us two points – the upper-right vertex A and the lower-right vertex B.

To find the optimal value for x that minimizes the length of the fold (y), we need to consider the properties of right triangles.

By folding the upper-right corner, we create a right triangle ABC, where A is the vertex of the fold, B is the lower-right vertex, and C is the point where the fold intersects the bottom edge as shown below:

8 in.
←─────────┐
│ │
x│ │
│ A │
│─────────→
│ C │
│ │
│ │
└─────────┘

Now, we can use the properties of right triangles to determine the relationship between x, y, and the dimensions of the paper.

In a right triangle, the Pythagorean theorem states that the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. In our case, the legs of the right triangle are x and (8-x), and the hypotenuse is y.

Applying the Pythagorean theorem, we get:
x^2 + (8-x)^2 = y^2

To minimize y, we need to find the value of x that minimizes the expression for y. To do this, we can find the minimum point by taking the derivative of the expression with respect to x and setting it equal to zero:

d/dx (x^2 + (8-x)^2) = 0

Simplifying this equation, we get:
2x - 2(8-x) = 0
2x - 16 + 2x = 0
4x - 16 = 0
4x = 16
x = 4

Therefore, to minimize the length of the fold (y), we need to fold the upper right-hand corner of the paper such that the distance from the right edge to the fold (x) is 4 inches.

To minimize the length of the fold, we need to find the length that results in the shortest distance between the upper right-hand corner and the bottom edge.

Let's denote the length of the fold as "x," and the resulting distance between the upper right-hand corner and the bottom edge as "y."

First, let's visualize the situation. We have a rectangle with dimensions 12 in. by 8 in. We want to fold the upper right-hand corner down to the bottom edge. To do this, we need to create a right triangle with one side being the fold (x) and the other two sides being the edges of the paper.

To find the length of y, we can use the Pythagorean theorem:

y^2 = (8 - x)^2 + (12 - x)^2

To minimize y, we need to find the value of x that minimizes the expression y^2. As y is positive, we take the square root of y^2 to find y.

y = √(8 - x)^2 + (12 - x)^2

To find the minimum value of y, we can take the derivative of y with respect to x, set it equal to zero (indicating a critical point), and solve for x.

dy/dx = (-2(8 - x) - 2(12 - x)) / 2√((8 - x)^2 + (12 - x)^2) = 0

Simplifying the derivative:

-16 + 2x - 24 + 2x = 0

4x = 40

x = 10

So, to minimize the length of the fold (y), we would fold the upper right-hand corner to the bottom edge at a distance of 10 inches from the upper right-hand corner.